AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/3/3-15)
Unit 3 · Topic 3.15
3.15 Rates of Change in Polar Functions
For a polar function r = f(θ), whether the point moves toward or away from the origin depends on both the sign of r and whether r is increasing or decreasing. Average rates of change of r with respect to θ measure how fast the signed radius changes per radian, and help you estimate values.
Key terms
- average rate of change
- distance from the origin
- relative extremum
- radius per radian
Distance from the origin
The distance from the origin to the point (r, θ) is |r|. So what matters is whether |r| is growing or shrinking.
| r is… | and is… | Distance from origin |
|---|---|---|
| Positive | Increasing | Increasing |
| Positive | Decreasing | Decreasing |
| Negative | Decreasing | Increasing |
| Negative | Increasing | Decreasing |
Why negative r flips the rule
If r is negative and decreasing, it goes from, say, −1 to −2 to −3. The values are getting smaller, but their sizes are getting bigger, so the point moves away from the origin.
If r is negative and increasing, from −3 to −2 to −1, the point moves toward the origin.
The quick test: the point moves away when r and its change have the same sign, and moves closer when they have opposite signs.
Using the graph of r against θ
When r = f(θ) is graphed on rectangular axes, the distance from the origin is how far the graph is from the horizontal θ-axis. So the polar point moves away from the origin exactly where the rectangular graph moves away from the θ-axis, whether above or below it, and moves closer where the graph heads toward the axis.
Closest and farthest points
When r switches direction on an interval of θ (rising then falling, or falling then rising), r has a relative maximum or minimum there, and the matching point on the curve is locally the closest to or farthest from the origin. Which one depends on the sign of r: a relative maximum of a negative r, like −1 between values near −3, is a close point, not a far one.
For example, r = 3 + 2 sin θ has a maximum value of 5 at θ = π/2, so the point (5, π/2) is the farthest the curve gets from the origin. Its minimum value of 1 at θ = 3π/2 gives the closest point.
Average rate of change of r
The average rate of change of r with respect to θ on [θ₁, θ₂] is (f(θ₂) − f(θ₁))/(θ₂ − θ₁). Its units are radius units per radian. It tells you how fast the signed radius changes per radian, on average.
You can use it to estimate values in between: f(θ) ≈ f(θ₁) + (average rate)(θ − θ₁). This is the same idea as using a secant line to estimate, as in topic 1.2.
A positive average rate means r increased overall on the interval, and a negative one means r decreased. On its own, that doesn't tell you whether the curve moved toward or away from the origin; you also need the sign of r.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Toward or away from the origin
For r = 1 + 2 cos θ, decide whether the point on the curve is moving toward or away from the origin on π/2 < θ < 2π/3 and on 2π/3 < θ < π.
Show the solutionHide the solution
- Step 1: On π/2 < θ < π, cos θ decreases, so r = 1 + 2 cos θ decreases: from r = 1 at π/2, to 0 at 2π/3, to −1 at π.
- Step 2: On π/2 < θ < 2π/3, r is positive and decreasing, so the distance |r| shrinks: the point moves toward the origin.
- Step 3: On 2π/3 < θ < π, r is negative and decreasing, so |r| grows from 0 to 1: the point moves away from the origin.
Answer: Toward the origin on π/2 < θ < 2π/3; away from the origin on 2π/3 < θ < π.
- Example 2Calculator allowed
Average rate of change and an estimate
For r = 2 + 3 sin θ, find the average rate of change of r on [0, π/6], interpret it, and use it to estimate r at θ = π/12.
Show the solutionHide the solution
- Step 1: r(0) = 2 and r(π/6) = 2 + 3(1/2) = 3.5.
- Step 2: Average rate = (3.5 − 2)/(π/6 − 0) = 1.5 · 6/π = 9/π ≈ 2.865.
- Step 3: On average, the signed radius increases by about 2.865 units per radian on this interval. Since r is positive and increasing, the curve moves away from the origin.
- Step 4: Estimate: r(π/12) ≈ 2 + (9/π)(π/12) = 2 + 9/12 = 2.75. (The actual value, 2 + 3 sin(π/12), is about 2.776.)
Answer: 9/π ≈ 2.865 radius units per radian; r(π/12) ≈ 2.75.
- Example 3
Trap: decreasing r, growing distance
A polar function has f(1) = −0.5, f(1.5) = −1.2 and f(2) = −2.0, and f is decreasing on 1 < θ < 2. A student says the curve gets closer to the origin there because r is decreasing. Is that right?
Show the solutionHide the solution
- Step 1: The distances from the origin are |r|: 0.5, 1.2 and 2.0. They are growing.
- Step 2: r is negative and decreasing, which is one of the two cases where the distance increases.
- Step 3: “Decreasing” describes the signed value of r, not its size.
Answer: No. The point moves away from the origin, because r is negative and decreasing.
Common mistakes
- Treating “r is decreasing” as “getting closer.” Check the sign of r too.
- Forgetting units for the average rate of change: radius units per radian.
- Finding the farthest point using r instead of |r|. A very negative r can be the farthest point.
On the exam
- Expect questions that give a polar function, or its graph on rectangular axes, and ask on which interval the curve moves toward or away from the origin. Justify with both the sign of r and whether r increases or decreases.
- Polar questions appear mainly in multiple choice; the free-response questions on unit 3 center on sinusoidal models and trig equations. Expect to find an average rate of change of r, give its units, or use it to estimate a value.
Connected topics
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Check yourself
4 questions on 3.15 Rates of Change in Polar Functions. Pick an answer to see if you got it, and why.
The polar function r = g(θ) is given by g(θ) = 2 + sin(3θ). Which of the following is closest to the average rate of change of r with respect to θ on the interval 0.5 ≤ θ ≤ 1?
For the polar function r = 3 sin(2θ), on which of the following intervals is the distance between the point (r, θ) and the origin increasing?
The polar function r = f(θ) is given by f(θ) = 1 + 2 cos θ for 0 ≤ θ ≤ 2π.
Which of the following is true for 2π/3 < θ < π?
What is the average rate of change of r with respect to θ over the interval 0 ≤ θ ≤ π/2?
0 of 4 answered